By Ned Chapin

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**Additional info for An introduction to automatic computers**

**Sample text**

The slider and duster are modeled as point masses. Our objective is to 1. Obtain the general equations of motion of a two-link manipulator To obtain the Euler–Lagrange equations, we must obtain the total kinetic and potential energies in terms of the total mass, moment of inertia and mass moment components. The slider and duster masses are Ms and M, respectively; mi and Li are, respectively, the ith link mass and the ith link length; Licg is the position of the CM of the ith link with reference to the ith joint and kicg is the ith link’s radius of gyration about its CM.

327) The total kinetic energy of the first three links and the mass of link four is T3 = ( ) )( ( 1 1 m2 L22 cg + m3 L22 + m4 L22 + I yy 2 q22 + m3 L23cg + m4 L23 + I yy 3 q2 + q3 2 2 + { ( ) ( ) 2 } ) 1 I zz1 + m2 L22 cg + m3 L22 + m4 L22 + I yy 2 cos2 q2 + m3 L23cg + m4 L23 + I yy 3 cos2 ( q3 + q2 ) q12 2 ( ) + ( m3 L2 L3cg + m4 L2 L3 ) cos q2 cos ( q3 + q2 ) q12 + ( m3 L2 L3cg + m4 L2 L3 ) q2 q2 + q3 cos q3 . 328) The rotational kinetic energy of the fourth link is T4 = ( ( 1 I xx 4 cos q4 cos ( q2 + q3 ) q1 - sin q4 q2 + q3 2 ( ( )) 1 + I yy 4 sin q4 cos ( q2 + q3 ) q1 + cos q4 q2 + q3 2 2 )) 2 2 1 + I zz 4 q4 - sin ( q2 + q3 ) q1 .

The length of the first link is L and the distance of the CG of the second telescoping link from the end of the first link is d. 264) The translation kinetic energy is T1 = 1 2 1 1 1 å m {( x ) + ( y ) + ( z ) } = 2 m L ( q + f ) + 2 m ( L + d ) ( q + f ) + 2 m d . 266) êë W z úû êë rB úû êë0 úû êë coss q úû The moment of inertia of the capstan about its axis of rotation is I1. Its angular velocity is f. 273) ¶q ( ) The Euler–Lagrange equations are (m L 2 1 cg 2 1 cg dding a point A mass at the tip 2 - 2 ( I 2 + I 3 ) cos q sin qfq = t1, (m L ) + m2 ( L + d ) + ( I 2 + I 3 ) cos2 q + I1 f + 2m2 ( L + d ) df ) + m2 ( L + d ) + I 2 + I 3 q + 2m2 ( L + d ) dq + ( I 2 + I 3 ) j2 sin q cos q 2 ( ) + g m1Lcg + m2 ( L + d ) cos q = t2 .